A particle A and its antiparticle B are travelling in opposite senses in free circular orbits in the equatorial plane of the Schwarzschild geometry, one at coordinate radius rA and the other at rB(>rA). At some instant, A emits a photon of frequency νA that travels radially and is received by B with frequency νB. Show that
νAνB=(1−3μ/rB1−3μ/rA)1/2.
Suppose now that rA=rB=r, so that the particles collide and annihilate each other. Show that the total radiated energy as measured by an observer at rest at the point of collision is given by
E=2m0c2(1−3μ/r1−2μ/r)1/2,
where m0 is the rest mass of each particle.